I am curious about two questions below
Let $M$, $N$ be two topological manifold.
If $\dim M>\dim N$, is there exist an injective continuous map $f: M\rightarrow N$?
If $\dim M<\dim N$, is there exist an surjective continuous map $f: M\rightarrow N$?
To the first question, I think we can construct a map $F: M\rightarrow N\times\mathbb R^{m-n}$, which satisfies $F(p)=(p,0)$. If $f$ is injective, then $F$ is injective too. Then from invariance of domain, we can get $F$ is an embedding. Also, $f$ is an embedding. So is there exist embedding from $M$ to $N$?
Second question can be transfered $N$ as $\mathbb R^n$ by using local coordination. Then I have know idea.
Any advice is helpful. Thank you.
For the first question, the answer is no, there is no continuous injective map from a larger manifold to a smaller one.
Suppose there existed such an $f$. Then restricting $f$ to a chart $U\subseteq M$ we get an injective map $f|_U:U\rightarrow N$. By further restricting $U$, we may assume $f(U)$ is a subset of a chart $V$ on $N$.
Thus, by composing with chart maps (which are homeomorphisms, and therefore injective), we may assume $f$ maps an open subset of $\mathbb{R}^m$ to a subset of $\mathbb{R}^n$ (with $m = \dim M$ and $N = \dim n$). Abusing notation, I'll still write $U$ for this open subset of $\mathbb{R}^m$.
Now, consider the map $g:U\subseteq \mathbb{R}^m$ to $\mathbb{R}^m$ given by $g(x) = (f(x), \vec{0})$ with $\vec{0}\in\mathbb{R}^{m-n}$. This will also be injective with image a subset of $\mathbb{R}^n\times \{\vec{0}\}$. By Invariance of Domain, $g(U)$ is an open subset of $\mathbb{R}^m$. But every open subset which contains a point of the form $(f(x), 0)$ also contains a point of the form $(f(x), v)$ for some tiny $v\in \mathbb{R}^{m-n}$. This contradiction implies no such $f$ exists.
For the second question, yes, there are examples. Perhaps the most famous are the Peano curves. With some technical details sorted out, I believe one can prove that every (connected) topological manifold is the continuous image of $\mathbb{R}$.